*T.£ 


-/ 


MATHEMATICAL  MICROBES" 


A    PAPER 


READ    FOR 


"THE  CLUB"   (LITERARY) 


SPRINGFIELD,   MASS. 


A.  R.  BUFFINGTON, 

AN   EX-MEMBER 


(PRINTED  BV  REQUEST  OF  THE  CLUB) 


"MATHEMATICAL  MICROBES" 


A   PAPER 


READ    FOR 


"THE  CLUB"  (LITERARY) 


—  OF  — 


SPRINGFIELD,  MASS. 


FEBRUARY  nth,  1898 


A.  R.  BUFFINGTON, 

AN  EX-MEMBER 


(PRINTED  BT  BEQUEST  OP  THE  CLUB) 


PRISS  OF  THE  IRON  ERA 
DOVER,  N.  J. 


Qft 


MATHEMATICAL  MICROBES. 


I  call  the  subject  of  this  paper  "  Mathematical  Mi- 
crobes,1' as  the  quantities  involved  are  microscopical, 
but  essential  to  exactness — as  essential  as  the  discovery 
of  microbes  to  correct  diagnosis  of  disease,  assuming 
that  every  disease  has  its  distinctive  microbe.  You  are 
to  judge  whether  it  is  a  diseased  mathematics  to  which  I 
call  your  attention  to-night. 

In  the  field  of  criticism,  it  seems  to  me  "  common 
sense  "  judgment  is  either  altogether  excluded  or  un- 
graciously tolerated.  By  "  common  sense,''  let  us  un- 
derstand that  intuitive  knowledge  by  which  men  gener- 
ally pronounce  judgment  on  subjects  and  things  in  the 
knowledge  of  which  they  have  not  been  specifically 
educated.  A  judgment  based  on  common  sense — on 
"  The  eternal  fitness  of  things" — does  not  frequently 
accord  with  the  technique  of  a  cult  or  a  science.  To  be 
specially  educated,  let  us  say,  to  skilfully  handle  the 
tools  of  those  who  have  gone  before  in  any  calling, 
ought  not  to  darken  the  discernment  of  any  truth  per- 
taining to  it.  The  wider  and  greater  the  knowledge 
the  more  tolerant  of  special  ignorance  it  should  be,  for 
the  wisest  is  but  a  child,  and  his  wisdom  often  more  a 
matter  of  chance  than  ability.  Chance  has  played  a 
greater  role  in  the  achievements  of  men  than  is  com- 
monly understood.  To  see  nothing  that  does  not  fit 
into  the  moulds  that  shaped  their  attainments  in  any 
branch  of  knowledge  may  not  apply  to  the  pioneers — 
the  originators,  the  high  priests,  whether  by  sheer 
ability  or  by  accident — of  knowledge,  but  it  seems  to 
stamp  the  merely  learned  in  what  other  men  have  ac- 
complished. It  is  these  who  shape  the  opinions  of 
less  learned  men  and  inoculate  them  with  intolerance. 


4  MATHEMATICAL    MICROBES. 

Now,  common  sense  may  be  truer  even  than  mathe- 
matics, and  when  combined  with  the  genius  chance, 
may  accomplish  what  mathematics  cannot  in  its  own 
proper  field  ;  that  is,  eradicate  error  and  establish 
truth.  Let  it  be  understood  that  you  owe  this  paper 
to  chance,  and  that  the  preceding  wise  remarks  are 
merely  to  pave  the  way  for  a  statement  of  how  it  hap- 
pened, and  what  followed.  As  a  student  I  accepted 
my  mathematics  as  I  accepted  my  religious  faith — both, 
as  presented  to  me  by  books  and  teachers,  accorded 
with  my  discernment  of  truth.  I  no  more  thought  of 
questioning  one  than  the  other  and  in  this  desirable 
frame  of  mind,  so  far  as  mathematics  is  concerned,  I 
continued  until  quite  recently. 

Two  or  three  years  ago  I  had,  in  some  of  my  me- 
chanical work,  evolved  a  peculiar  triangle  whose  solu- 
tion I  deemed  it  desirable  to  have,  and  I  set  about  it  in 
the  ordinary  way,  but  this  way  proved  unsatisfactory, 
as  it  involved  the  extraction  of  the  square  root  of  im- 
perfect squares.  I  wanted  perfect  values,  not  approx- 
imations, but  to  get  them  I  needed  relations,  or  equa- 
tions, not  based  on  the  Pythagorean  law  that  the  square 
of  the  hypothenuse  of  a  right  angle  triangle  equals  the 
sum  of  the  squares  of  the  other  two  sides.  To  avoid 
the  square  root,  I  required  an  equation  involving  but 
the  first  power  of  the  variable,  or  unknown  quantity. 
Such  an  one  would,  of  course,  give  values  freed  from 
the  inaccuracies  of  the  square  root  of  imperfect  squares. 
In  my  efforts  to  obtain  this  I  discovered  that  in  sub- 
tracting an  algebraic  expression  from  one  member  of 
an  equation  I  had  evolved,  there  was  no  difference  in 
result  ;  that  is,  whether  it  were  subtracted  or  not  the 
resulting  values  were  not  affected.  Now,  there  could 
be  but  one  meaning  for  such  a  result,  and  that  was, 
that  this  algebraic  expression  must  be  nothing,  must 
be  zero,  for  only  zero  subtracted  from  a  thing  could 
give  a  difference  equal  to  the  thing  itself.  It,  the  said 
expression,  contained  but  the  first  power  of  the 
unknown  quantity,  and,  as  explained,  being  zero,  if 
equated  with  zero, — it's  equal, — would  furnish  the  equa- 


MATHEMATICAL    MICROBES.  5 

tion  I  was  seeking.  Accident  had,  apparently,  given 
me  what  I  had  failed  to  work  out  and  I  thus  found  the 
values  sought  without  extracting  the  square  root  of 
imperfect  squares.  But  I  was  confronted  with  some- 
thing that  shook  my  mathematical  faith  to  the  founda- 
tion, viz  :  if  these  values  were  true  the  Pythagorean 
law  was  not  for  the  particular  triangle  involved.  How 
could  this  be  so,  for  the  proof — geometrical  proof — of 
that  law  is  general,  not  particular  ?  It  applies  to  any 
right  angle  triangle,  how  then  can  there  be  an  excep- 
tion ?  In  the  face  of  so  perfect  a  proof  who  would  dare 
proclaim  an  exception  ?  The  old  apothegm  that  "  The 
exception  proves  the  rule"  would  not  apply  in  this  case, 
and  I  should  have  to  establish  it — the  exception — by 
well  known  and  accepted  mathematical  methods  before 
it  would  be  received  by  any  one.  This  I  found  to  be 
no  easy  task,  but  I  labored  at  it  faithfully  and  have 
now  arrived  at  a  demonstration  satisfactory,  at  least, 
to  myself. 

In  the  Atlantic  Monthly  for  August,  1897,  is  a  paper 
entitled,  "  The  pause  in  criticism — and  after,"  in  which 
occurs  this  statement  : 

"  Every  so-called  law  was  originally  only  the  opin- 
"  ion  of  one  man." 

Again.  "  When  knowledge  has  reached  the  stage 
"  where  it  can  be  packed  into  formulas  one  of  two 
"  things  happens  :  either  the  formulas  are  easily  learned 
"and  repeated  mechanically,  which  leads  to  petrifac- 
"  tion,  or  they  serve  as  new  points  of  departure  from 
"  which  the  untrammelled  "  mind  "  sets  out  on  a  higher 
"  quest." 

As  an  illustration  of  the  former  case  rhetoric  is 
used,  and  then  the  paper  adds  :  "  How  different  is  the 
"  aspect  of  those  sciences  and  arts  in  which  ciassifica- 
"  tion  neither  implies  arrested  development  nor  marks 
"the  limit  beyond  which  progress  cannot  be  made  ! 
"  We  need  cite  as  an  illustration  only  the  mathematics, 
"  one  of  the  branches  of  knowledge  in  which  fixed  laws 
"  were  earliest  formulated,  and  the  science  above  all 
"  others  in  which  absolute  accuracy  can  be  attained  at 


6  MATHEMATICAL    MICROBES. 

"  every  step  ;  age,  for  it,  does  not  mean  senility  ;  rules 
"  are  not  shackles.'1  To  this  the  majority  will  agree, 
not  only  as  to  the  present,  but  the  past  aspect  of  math- 
ematics. Yet  even  this  may  fall  within  the  domain  of 
the  criticism  ;  that  is,  it  may  not  be  exempt  from  class- 
ification with  the  inexact  sciences  packed  into  formulas, 
easily  learned  and  repeated  mechanically,  not  only 
leading  to  petrifaction,  but  constituting  completed  forms 
of  petrifaction. 

To  a  more  or  less  limited  extent  the  old  aphorism 
that  "Truth  is  many  sided''  may  apply  even  to  mathe- 
matics. Among  its  formulas  there  may  be  some  that 
comprehend  but  one  face  of  truth,  rejecting  all  others. 

In  these  latter  days  of  a  marvelous  century  there 
are  evidences,  not  hard  to  find,  that  man  has,  to  a  large 
extent,  freed  himself  from  the  shackles  of  environment, 
education  and  the  consequent  prejudices,  sufficiently 
even  to  listen  with  some  degree  of  patience  to  question- 
ings of  mathematical  truth.  Knowledge  grows  by 
ever  widening  circles.  Time  has  long  since  passed,  when 
a  scientist,  so  called,  was  "a  jack  of  all  trades"  and 
perfect  in  all;  when  a  college  professor  occupied  several 
chairs  and  felt  competent  to  instruct  in  any  branch  of 
knowledge. 

So  vast  has  become  the  field  in  which  the  searchers 
for  truth  are  working  that  it  may  be  said,  without  ex- 
aggeration, that  man's  short  life  may  be  more  than 
filled  by  the  study  of  a  single  drop  of  water.  Not  only 
new  fields  are  entered  but  old  ones  are  gleaned  over 
for  grains  of  truth  that  may  have  escaped  the  first 
reapers. 

It  is  into  one  corner  of  one  of  these  old  fields  I  wish 
to  take  you  for  a  search  after,  perchance,  a  lost  grain, 
overlooked — hidden — in  the  long  ages  since  Pythagoras 
announced  the  law  I  have  stated, — which  is  the  very 
corner  stone  of  mathematics.  This  law,  ever  since 
enunciated,  has  been  accepted  by  "  all  sorts  and  condi- 
tions of  men"  as  an  unchanging  and  unchangeable 
truth,  absolutely  without  an  exception.  To  question 
its  universality — to  say  that  it  has  exceptions — would 


MATHEMATICAL    MICROBES.  7 

appear  to  be,  prima  facie,  evidence  of    an   unbalanced 
mind,  or,  of  a  temerity  bordering  on  the  marvelous. 

While  accepting  the  positive  value  of  the  proof — in 
fact,  presenting  as  I  shall  myself,  a  proof  more  positive 
and  convincing  if  possible  than  the  proofs  found  in  the 
books — I  have  mathematical  ground  for  questioning 
the  universality  of  the  law,  and  in  placing  it  before  you 
I  beg  to  say  that  it  involves  no  abstruse  paths  hard  to 
follow.  There  is  nothing  in  it  beyond  the  four  rules  of 
arithmetic  and  the  extraction  of  the  square  root — noth- 
ing but  what  any  school  boy  can  follow  and  what  he 
may  read  as  he  runs.  Yet,  the  hardest  to  prove  is  the 
simplest  truth!  One  truth  ends  where  another,  which 
it  cannot  include,  begins.  This  may  be  the  basis  of  the 
saying  that  "  the  exception  proves  the  rule.'1  A  law  or 
rule  to  be  universal  can  have  no  exception  and  the  law 
of  Pythagoras,  as  proved,  admits  of  no  exception,  it  is 
therefore  universal  if  the  proof  be  admitted.  That  the 
square  of  the  hypothenuse  of  a  right  triangle  equals  the 
sum  of  the  squares  of  the  other  two  sides  is  a  law,  and 
one  truth,  does  not  annul  other  truths.  The  mere  state- 
ment of  it  involves  another  law,  and  states  another  law 
which  it  cannot  make  void  should  it  in  any  case  conflict 
with  it.  Both  laws  must  stand  inviolate  with  reference 
to  each  other.  To  say  that  one  thing  equals  another, 
which  the  Pythagorean  law  does,  means  an  equation — 
states  an  equation — the  members  of  which  are  inter- 
changeable and  absolutely  equal.  Whatever  one  is,  the 
other  either  must  be,  or  readily,  from  the  existing 
equality  of  values,  be  convertible  into  it.  a  =  b  is  an 
equation  and  when  the  values  of  the  letters  be  substi- 
tuted for  them,  one  member  becomes  but  a  repetition 
of  the  other.  If  "a"  expresses  in  algebraic  terms 
a  geometrical  figure,  "£"  must  be  the  same, 
or  the  equal  in-  area  of  this  figure,  and  when 
this  area  is  expressed  in  numbers  of  some  unit 
or  subdivisions  of  it,  the  area  of  "  a,"  whatever  be 
the  figure  represented,  must  be  the  same.  If  "a"  rep- 
resent a  perfect  square,  geometrically  and  algebraically, 
"  £,"  however  expressed,  must  also  be  the  equal  in  area 


8  MATHEMATICAL    MICROBES. 

of  this  perfect  square  and  geometrically  convertible  in- 
to it.  In  short,  one  member  of  an  equation  cannot  be 
even  infinitesimally  less  or  greater  than  the  other  ;  also 
if  one  be  a  positive,  determinable  figure  or  quantity  the 
other  must  be  the  same — one  cannot  be  a  transcenden- 
tal, incommensurable  or  indefinite  quantity  and  the 
other  a  positive,  definite  and  commensurable  one  with- 
out making  void  the  truth  the  equation  stands  for,  or 
expresses.  This  is  what  appears  to  take  place  when  the 
Pythagorean  law  is  made  universal ;  one  face  of  truth 
is  recognized  to  the  prejudice  of  another  equally  as  im- 
portant and  equally  as  powerful  for  the  establishment 
of  corr«ct  knowledge — for  the  maintenance  of  that  ex- 
actness without  which  mathematics  becomes  unreliable. 
This,  I  take  it,  is  common  sense — is  a  common  sense 
judgment.  That  any  two  perfect  squares  added  together 
will  produce  a  perfect  square,  in  the  nature  of  things 
strikes  one  as  a  statement  that  must  have  a  proof  devoid 
of  any  weak  point.  Now,  when  we  leave  the  geometri- 
cal proof — which,  so  far  as  I  know  or  any  other  person 
known^  to  me  knows,  has  no  weak  point — and  sub- 
^stitute, things  themselves  ^^ftheir  respective  symbols} 
we  must  be  satisfied — except  in  all  cases  involving 
what  have  been  rightly  called  "right  angle  triangle 
numbers" — with  equations  one  member  of  each  of 
which  is  covered  by  the  "  radical  sign,"  meaning  the 
extraction  of  the  square  root.  The  square  roots  of 
the  first  members  of  these  equations  have  been  ex- 
tracted :  they  were  perfect  squares,  but  their  conceded 
equals,  now  under  the  radical  sign,  are  not  perfect 
squares,  tho'  members  of  equations,  proclaiming  perfect, 
not  approximate,  equality  with  perfect  squares.  It 
strikes  me  there  must  be  something  wrong  with  this 
state  of  things.  Symbols,  to  be  symbols  at  all,  must 
equal  the  things  represented  or  they  are  false.  To  say 
let  x,  or  y,  or  2,  represent  this  line  or  that  line,  or  this 
square  or  rectangle,  or  that  square  or  rectangle, 
means  that  the  person  who  says  so  stipulates  that  in  all 
he  proposes  to  do  with  x,y,orz;  x,y,  or  z  shall,  not  may, 
truthfully  and  unequivocally  represent  the  said  line, 


MATHEMATICAL    MICROBES.  9 

square,  or  rectangle.  It  is  convenient,  simplifies,  and 
saves  labor  for  him  or  for  those  he  wishes  to  inform, 
or  argue  with,  to  use  symbols  for  the  things  them- 
selves, but  if  he  be  held  to  a  strict  account  in  the  use 
of  them,  he  cannot  produce  results  that  would  not  have 
been  arrived  at  had  the  things  themselves  been  used. 
If  he  does,  then  we  have  a  right  to  question  the  truth 
of  the  results — he  has  not  kept  faith  with  his  own  agree- 
ment. Ah,  but  mathematicians  will  say,  the  things 
themselves  are  not  commensurable — there  is  no  common 
unit,  hence  the  approximation  of  the  square  root.  Num- 
bers are  not  flexible,  not  divisible,  and  cannot  be  used 
where  their  symbols  can  be ;  cannot  be  made  to  follow 
their  symbols  thro'  the  labyrinth  of  mathematical  pro- 
cesses— algebra  is  not  arithmetic.  Arithmetic  is  well 
enough,  but  it  is  limited ;  when  we  go  into  abstruse, 
complicated  mathematical  processes  numbers  cannot 
be  used,  we  must  have  some  other  tools.  These  enable 
us  to  make  algebraic  squares  of  the  first  members  of 
the  equations  you  are  criticising.  In  the  untrammelled 
region  of  mind — the  boundless  field  wherein  man's 
mind  may  work  in  abstruse  speculation  or  mathe- 
matical work — realities,  numbers,  have  no  place.  These 
would  clog  the  work;  no  useful  results  could  be  pro- 
duced. This  sounds  like  good  logic ;  it  satisfies  the 
mathematician  and  he  makes  it  satisfy  other  men.  The 
syllogisms  are  all  right  if  the  premises  be  granted,  but 
the  premises  I  challenge.  The  things  themselves  are 
not  commensurable,  it  is  said,  if  approximations  be  pro- 
duced by  combining  them.  There  never  was,  it  seems 
to  me,  either  actually  visible,  tangible  and  existing,  or 
conceived  of  in  the  "  boundless  field  wherein  man's 
mind  may  work,"  a  line,  surface  or  solid,  that  is  not 
commensurable  with  any  other  line,  surface,  or  solid  so 
existing  or  conceived  of,  because  they  are  all  infinitely 
divisible.  There  is  no  limit  to  the  division  of  the 
things  compared  and  it  can  be  carried  on  until  a 
common  unit  can  be  found  ;  when  that  is  reached  the 
things  are  commensurable.  This  statement,  or  premise, 
therefore,  amounts  to  nothing  more  than  this,  "  we  get 


10  MATHEMATICAL    MICROBES. 

these  incomplete  squares  for  the  second  members  of 
the  equations  referred  to  because  we  have  not  a 
common  unit  for  the  lines,  &c.,  dealt  with."  Thus  the 
mathematician  has  not  been  able  to  find  a  common  unit 
and  has  been  obliged  to  make  one  for  himself — the 
decimal — with  which  he  makes  the  world  to  square. 
When  he  finds  two  things  incommensurable — that  is, 
not  commensurable  within  the  method  used  for  extract- 
ing the  square  root,  he  applies  to  the  problem  the 
decimal  division  of  the  unit.  So  poor  is  he  that  he  has 
only  one  out  of  the  infinite  number  into  which  the  unit 
may  be  divided.  This  brings  me  to  the  second  state- 
ment or  premise,  viz.:  that  numbers  are  not  flexible 
and  cannot  be  made  to  follow  their  symbols  in  laby- 
rinthian  mathematical  processes.  This  statement  has 
also  been  accepted — has  been  granted  to  the  mathe- 
matician. Numbers  and  the  units  of  numbers  are  as 
flexible  as  their  symbols  can  be.  They  can  be  carried 
thro'  as  many  and  complicated  mathematical  gymnas- 
tics as  their  symbols  or  representatives  ;  no  more,  no 
less.  They  are  infinitely  divisible  and  their  units  are 
infinitely  divisible  ;  so  much  and  no  more  can  be  said 
of  their  symbols.  Besides  symbols — representatives — 
if  they  are  capable  of  but  one  mathematical  summer- 
sault beyond  what  the  things  represented  are  capable 
of,  they  cease  to  be  symbols  or  representatives  of  these 
things. 

With  so-called  square  roots,  and  the  method  of 
obtaining  them  I  have  no  quarrel.  The  square  root, 
if  imperfect,  approximates  to  something  that  does  not 
exist.  It's  the  nearest  approach  that  man  has  yet  made 
to  a  creation.  He  produces  by  algebraic  process  some- 
thing, or  thinks  he  does,  he  calls  a  square,  which  is  not 
a  square,  because  there  is  nothing  in  nature  that  will 
represent  or  express  a  side  of  it,  and  then  gravely  pro- 
ceeds to  extract  its  square  root,  which  on  inspection  is 
found  as  mythical  as  his  square.  He  cannot  construct 
his  square  and  should  he  live  forever  he  could  not  reach 
in  his  process — tho'  it  be  scientific — a  completed  line  to 
represent  a  side  of  it.  Both  of  them — square  and  line 


MATHEMATICAL    MICROBES.  II 

— are  the  purest  fiction  and  this  is  called  mathematics  ! 
That  a  given  area — note  that  I  say  given,  that  is, 
actually  existing — can  be  represented  approximately 
by  one  of  the  infinite  number  of  squares  that  can  be 
made  from  the  infinite  number  of  sides  that  can  be 
produced  by  extracting  the  square  root  of  the  number 
expressing  the  area,  is  a  truth  so  far  as  it  goes,  but  we 
know  what  it  is  when  we  get  it  and  we  know  how  we 
got  it.  There  is  no  questionable  point  in  the  process. 
It  is  an  approximation  ;  it's  what  we  started  out  to 
get,  and  it  is  limited  only  by  the  number  of  decimals 
at  which  we  stop  in  the  process  of  extracting  the  square 
root.  The  process  itself  is  not  only  scientific  but  it  is  a 
beautiful  one — one  that  enables  us  readily  to  get  the 
square  root  of  perfect  squares,  as  well  as  approxima- 
tions for  unperfect  squares. 

The  existing  algebra,  trigonometry,  and  even  the 
differential  calculus  are  permeated  with  the  fictions, 
the  method  of  the  production  of  which  I  have  pointed 
out.  It  is  true  these  fictions  are  approximations  and 
differ  from  the  true  values  by  microscopical  quantities, 
and  for  all  ordinary  purposes  satisfy  requirements. 
For  astronomical  calculation  involving  inconceivable 
distances  they  might  make  appreciable  differences. 

In  a  paper  of  this  kind  I  can  but  give  a  mere  outline 
of  the  detailed  proof  of  the  existence  of  the  small  quan- 
titives  that  have  been  neglected.  Before  doing  so  I 
will  give  a  proof  of  the  Pythagorean  law  so  concise  and 
convincing  that  there  would  appear  to  be  no  escape 
from  the  conclusion. 


12  MATHEMATICAL    MICROBES. 

Observe  this  triangle — right  angle  triangle: 


It  is  the  geometrical  embodiment  of  the  algebraic 
statement  that  the  sum  of  the  squares  of  two  quantities 
equals  the  sum  of  the  square  of  their  difference  and 
twice  their  rectangle.  Thus  a"+b2  =  (a— b)a+2ab.  This 
simple  algebraic  law  is  only  another  statement  of  the 
Pythagorian,  and  is  its  perfect  proof.  You  see  in  the 
figure  the  square  of  the  difference  of  the  base  and  per- 
pendicular surrounded — symmetrically  and  completely 
filling  the  remaining  area  of  the  square  on  the  hypothe- 
nuse — by  four  right  angle  triangles  exactly  equal  to  the 
given  one  and  the  sum  of  them  equal  to  twice  the  rect- 
angle of  the  base  and  perpendicular  of  it — the  whole 
together  being  the  square  on  the  hypothenuse,  which 
is  therefore  equal  to  the  sum  of  the  squares  of  the  other 
two  sides  of  the  given  triangle.  Now  if  you  give  these 
sides  values  you  can  place  the  values  of  the  parts,  into 
which  the  square  on  the  hypothenuse  is  divided,  on  them 
and  the  sum  proclaims  at  once  that  this  square  equals 
the  sum  of  the  squares  of  the  other  two  sides.  Call  the 


MATHEMATICAL    MICROBES.  13 

base  3  and  the  perpendicular  £,  the  difference  is  2^,  the 
square  of  which  is  6£.  The  area  of  the  given  triangle 
is  3X^-f-2  =  f.  Four  times  f  is  3,  which  equals  twice  the 
rectangle  of  the  two  sides.  Thus  the  eye  as  well  as  the 
understanding  takes  in  the  truth  of  the  Pythagorean 
law. 

The  particular  triangle  I  have  mentioned  as  resulting 
from  some  work  in  which  I  was  engaged,  was  peculiar 
in  that  but  one  side — the  base — and  part  of  the  hypothe- 
nuse  were  known,  with  which  to  determine  the  other 
parts. 

Observe  this  figure  (Fig.  2)  which  presents  this  tri- 
angle: 


FIQ.  2. 


The  figure  shows  the  square  erected  on  the  hypothe- 
nuse  of  the  triangle.  The  base  ac  is  3  and  that  part  of 
the  hypothenusef  bd,  outside  of  the  circle  described  on 
the  base,  is  i.  The  line,  cb,  we  know,  from  one  of  the 
properties  of  the  circle,  is  perpendicular  to  the  hypothe- 
nuse  at  the  point  £,  and  extended  up  thro'  the  square 
on  the  hypothenuse,  it  divides  it  into  two  rectangles,  the 
larger  one,  by  the  Pythagorean  law,  being  equal  to  the 


14  MAMHEMATICAL    MICROBES. 

square  on  the  base;  the  smaller,  equal  to  the  square  on 
the  perpendicular.  The  square  erected  on  ab,  the  un- 
known part  of  the  hypothenuse,  divides  the  larger  rect- 
angle into  two  parts  —  a  perfect  square  and  a  rectangle, 
the  side/tf  '  of  the  latter  being  equal  to  bd,  the  known  part 
of  the  hypothenuse,  and  if  the  upper  half  of  it,  a'efb  ', 
be  removed  and  placed  in  the  position,  bihg,  it  will  fill 
half  the  space  gbdq.  Thus  the  whole  of  the  larger  rect- 
angle will  become  the  irregular  figure  efghiba  —  that 
is,  a  square,  lacking  but  a  corner,  which,  if  filled  in,  will 
make  it  perfect,  and  we  see  that  this  corner  is  the  square 
of  bi_  —  half  of  bd.  We  have  thus  completed,  geometri- 
cally, the  square  of  the  larger  rectangle  and  if  it  equal 
the  square  on  the  base,  the  same  quantity  added  to  the 
square  on  the  base  will  make  the  sum  equal  to  the 
completed  square  of  the  larger  rectangle.  Now,  if  the 
base,  «£,  with  a  as  a  pivot,  be  swung  up  into  the  hypothe- 
nuse, it  will  take  the  position  ak  and  on  it  we  will  erect 
the  square  makl  —  in  other  words,  the  square  on  the 
base  ac^  has  been  transferred  to  a  new  position  on  the 
hypothenuse,  so  that  we  have  it  inside  of  the  completed 
square  of  the  larger  rectangle.  The  space  —  area  —  be- 
tween the  two,  outside  of  one  and  inside  of  the  other, 
must  equal  the  addition  fghj  —  the  square  of  bi  —  which 
we  added  to  the  larger  rectangle  to  complete  its  square, 
?y  the  square  on  the  base  be  equal  to  the  larger  rectangle. 
If  this  space  between  the  squares,  which  we  see  is  com- 
posed of  a  minute  square  and  two  small  rectangles,  be 
not  equal  to  the  square  of  bi,  the  Pythagorean  law  for 
this  particular  triangle  cannot  be  true. 

Having  given  the  geometry  of  the  matter  we  will 
now  go  into  the  algebra  of  it  and  between  the  former 
and  latter,  you  will  observe,  there  is  perfect  accordance, 
already  shown  in  the  proof  I  have  given  of  the  Pytha- 
gorean law,  according  to  which  (abY~\-(bcY=(acY=gt 
as  ab^  and  bc_  are  respectively  the  base  and  perpendicular 
of  the  right  angle  triangle  abc.  Let  us  call  #£,  x,  be 
will,  under  the  law,  be  a  mean  proportional  between  06 
and  bd;  that  is,  its  square  will  equal  abKbd,  and  &•  itself 


equal  \Tabxbd  —  \/  xXi  =  tfx.    Substituting  for  ( 


MATHEMATICAL    MICROBES.  15 

and  (be)*  their  symbals,  thus  assumed  and  determined, 
under  the  law,  we  have  the  equation 

*'+*=9-  •••(') 

apparently  at  first  the  only  equation  that  could  be  con- 

structed for  finding  the  values  of  the  unknown  parts  of 
the  given  triangle.  To  do  this  from  this  equation,  the 
first  member  x*-\-xy  altho'  the  equal  —  so  made  —  of  a  per- 
fect square  (9),  is  not  one  algebraically  and  must  be 
made  one.  This  is  done  by  taking  half  of  the  coefficient 
of  the  first  power  of  x,  squaring  it  and  adding  it.  This 
coefficient  is  i  (which  we  see  is  the  value  of  bd,  with 
which  we  multiplied  a&  to  get  an  expression  for  be),  half 
of  it,  is  y^  and  this  squared  is  %,  which  added  to  the'first 
member  of  equation  (i)  makes  it  x*-\-x-\-^.  This  addi- 
tion is  inadmissable  unless  the  same  be  made  to  the  sec- 
ond member,  altho'  it's  a  perfect  square  already  and  by 
the  law  of  the  equation  should  equal  an  area  in  the  first 
member  readily  convertible  into  a  perfect  square. 
The  equation  thus  becomes: 


and  we  are  confronted  with  the  same  state  of  things 
that  existed  in  the  equation  from  which  we  started. 
That  is,  we  have  one  member  a  perfect  square  whereas 
the  other  is  not,  but  in  the  last  case  the  responsibility 
is  shifted  from  algebra  to  arithmetic.  In  the  first  case 
arithmetic  was  equal  to  the  extraction  of  the  square 
root,  but  algebra  couldn't  do  its  part;  now  algebra  can, 
but  arithmetic  cannot.  However,  we  can  do  now,  what 
we  could  not  do  before,  we  can  force  a  solution- 
What  we  could  not  do  with  9'.?  equal,  that  is,  its  symbol 
or  representative,  we  can  make  p  itself  —  or  rather  0 
now  loaded  down  by  a  £—  do.  So  much  for  the  flexibil- 
ity of  symbols.  We  see,  however,  in  the  operation  the 
perfect  harmony  that  exists  between  algebra  and 
geometry  and  we  are  not  left  only  able  to  follow  the 
gymnastics  of  numbers,  or  things,;  in  mathematical 
work;  we  can  at  least,  thro'  the  eyes  of  geometry,  have 
some  idea  of  what  algebra  is  doing  and  possibly  in  that 
way,  keep  it  out  of  mischief.  Solving  the  last  equation 
by  extracting  the  square  root  of  both  members  we  arrive 


1 6  MATHEMATICAL    MICROBES 


+  to  infinity— 
a  value  very  much  resembling  a  tapeworm,  the  only 
essential  part  about  it  for  its  existence  being  its  head. 
The  tail  you  can  chop  off  at  any  place  suiting  your 
convenience  at  the  time,  but  the  head  remains  and  will 
live  even  if  deprived  of  its  last  decimal.  We  have 
found  something  for  the  second  member  of  a 
new  equation  which  we  call  the  value  of  x;  that  is 
of  the  side  ab  and  to  get  be  we  must  put  what 
is  left  of  this  poor  value  thro'  the  cruel  process  of 
going  again  thro'  the  decimal  shambles  to  come  out  an 
unrecognizable  abstraction  representing  nothing  in  this 
world  or  the  next;  the  x,  however,  and  the  \/xt  you  ob- 
serve, remain  unmutilated — only  the  despised  arithmetic 
suffers;  the  things,  not  the  symbols,  suffer.  Referring 
again  to  the  figure  2  we  are  concerned  with  the  two 
small  rectangles  and  the  minute  square,  to  which  atten- 
tion has  been  called.  In  whatever  way  it  is  attempted 
to  make,  or  find,  relations  between  these,  themselves, 
or  between  them  and  the  other  parts,  there  will  result 
either  the  expression  x*-\-x— ^  or  £  as  the  value  of  their 
sum;  showing  that  the  value  of  the  area  between  the 
square  9  and  the  completed  square  of  the  larger  rect- 
angle is  either  x*-\-x—zg-  or  \  depending  upon  how  it  is 
done.  If  x*-\-  x— ^-  be  equated  with  £,  to  which  the  law 
makes  it  equal,  it  results  in  returning  to  the  original 
equation  x*-{-x=<)  which  means  nothing  more  than  that 
we  have  forced  that  relation — it  does  not  mean  that  it  is 
true,  if  the  law  be  questioned.  An  examination  of  the 
method  by  which  the  £  is  determined  as  the  sum  of  the 
parts  of  the  area  between  the  squares  will  fix  the  parts 
of  which  this  \  is  the  sum.  The  small  line  ki  is  the  side 
of  the  minute  square  and  is  equal  to  ab+bi—ak;  that  is, 
x-\-%—3=x—%.  Squaring  this,  we  have  the  little  square, 
as  algebraically  expressed,  x*—$x-\-2£.  Multiplying  ki 
by  £/,  that  is,  x— f  by  3,  and  then  again  by  2  we  get  6x- 
15  for  the  algebraic  value  of  the  two  small  rectangles 
Now  if  in  the  expression  for  the  little  square,  the  value 
of  x*,  obtained  from  the  equation  x*-\-x=g,  that  is,  g—x, 
be  substituted  for  ,ra,  the  expression  becomes  V— 6*. 


MATHEMATICAL    MICROBES.  17 

Thus  we  obtain  the  algebraic  expressions  for  the  sum  of 
the  small  rectangles  and  the  little  square,  containing 
but  the  first  power  of  the  variable  x.  Adding  the  two 
together,  &£-—  6x-[-6x—  i5,the  sum  is  £,  but  examining 
the  two  expressions  we  see  that  this  would  be  the  sum 
no  matter  what  value  be  given  to  x\  that  this  sum  is 
entirely  independent  of  the  value  of  the  variable. 
Therefore  an  infinite  number  of  values  can  be  placed 
in  these  expressions  for  x  and  the  sum  of  them  will 
remain  the  same,  and  if  there  be  anything  wrong  with 
them  the  trouble  must  enter  into  the  value  (*£—6x), 
for  the  little  square  when  the  substitution  of  g—x  is 
made  for  x*.  _ 

Manifestly  if  ^—6x=x'1  —  ^x+^-,  the  \/&£-6x  must 


equal  x—  £.  We  have  an  equation  then,  ^^-—  6x=x—  f. 
Now  both  members  of  an  equation  can  be  multiplied 
by  the  same  quantity  without  affecting  their  equality. 
Multiply  both  members  of  this  equation  by  the  first,  it 


becomes  -^—  6x=(x—  f)  \/$£—6x.  Square  both  mem- 
bers to  get  rid  of  the  radical  and  we  finally  have 
(*£—  6xY=(x—  D'X^—  bx)  —  a  perfectly  regular  and 
legitimate  equation.  Squaring  the  parts,  as  indicated, 
and  reducing,  result  in  an  equation  of  the  third  degree, 
viz; 

24*'  =  37^+277^-549  (3) 

having  a  remarkable  property.  Two  values  of  the 
variable  will  satisfy  it,  viz;  x=y^l~^  and  x=%k,  the 
former  being  the  value  demanded  by  the  equation, 
xa-\-x=<),  the  latter  a  value  that  reduces  the  expression 
for  the  little  square  (after  substituting  in  it  g—x  for  x3) 
to  zero;  that  is,  *£•—  6x=o.  This  fact  requires  the  other 
part  entering  into  the  sum  \  —  the  part  expressing  the 
sum  of  the  two  little  rectangles  —  to  equal  £;  that  is, 
6^—15=^.  Hence  the  sum  6x—  15+^-—  bx=^,  as  de- 
manded by  the  condition  that  xa-\-x=gt  and  so  the 
space  between  the  square  9  and  the  completed  square 
of  the  larger  rectangle  oijf  the  square  on  the  hypothenuse 
must  equal  £,  as  demanded  by  the  Pythagorean  law! 
Again  taking  for  the  expression  ($£—6x)  for  the  little 
square,  its  equal  in  two  rectangles  on  two  sides  of  the 


l8  MATHEMATICAL    MICROBES. 

square  9,  or  in  one  twice  as  long  as  one  side;  that  is, 
dividing  the  expression  by  6,  the  sum  of  two  sides  of 
the  square  9,  we  have  the  quotient  61^|4-  as  one  of  the 
shorter  sides  of  an  equivalent  rectangle,  the  longer  of 
which  are  6.  Now  if  one  of  the  longer,  (6),  be  divided 
by  one  of  the  shorter  (&±:^A-)  the  quotient  will  be  the 
number  of  times  the  latter  is  contained  in  the  former, 
and  also  the  number  of  times  the  square  of  61^|4a;  is 
contained  in  the  area  of  the  equivalent  rectangle;  that 
is,  we  get  the  expression  -^J$%x,  which  represents  a  num- 
ber— a  ratio.  Therefore  the  square  of  6i^|*x  multiplied 
by  this  expression  will  equal  the  area  aforesaid  and  we 
have  this  equation 

(*^^Y<^&=^-  6x=**?*3-    ...     (4) 
a  glance  at  which  shows  it  to  be  a  true  one. 

Being  a  true  equation,  it  is  evident  that  were  the  ratio 
„  *  V.  known,  the  value  of  the  unknown  quantity  x 

o  1      £  4*  A  *' 

could  be  at  once  determined.  We  have  no  means  of 
knowing  this  value,  but  according  to  the  requirement 
of  the  Pythagorean  law,  which  in  this  case  is  that  x*-\-x 
shall  equal  9.  it  becomes  73_11424/?T.  Make  this  substi- 
tution, and  the  equation  becomes 

/  6  1-2  4  .A"         144         — 61-84JE 
V       84       /   13-1 2  V~5~T~         4 

Now  this  equation  gives  f£  for  the  value  of  x*  Not 
only  this,  but  any  value,  decimal,  fractional,  mixed,  or 
whole  number,  from  zero  to  infinity,  substituted  for 
this  ratio  will  give  the  same  value,  f£  for  x.  This 
could  not  be  possible  unless  f|—  x=o,  which  means  that 
-*•=-§£,  which  we  see  is  one  of  the  values  that  satisfies 
the  equation  "(3)''  of  the  third  degree,  already  discussed 
and  solves  the  riddle  of  the  sum  (^~ 6x-\-6x— 15)  of 
the  quantities  that  make  up  the  area  between  the  square 
9  and  the  larger  rectangle  of  the  hypothenuse  of  our 
triangle  after  having  its  "square  completed." 

I  have  gone  into  the  mathematical  details  as  little  as 
possible  to  make  the  subject  intelligible.  The  whole 
detailed  proof  of  what  I  have  outlined  to  you  I  have 
here  with  me  for  examination  if  desired.* 


*See  appendix,  a  careful  study  of  which  is  invited. 
^      £Y^*t-    >0~-' 


MATHEMATICAL    MICROBES.  19 

Than  those  I  have  touched  upon,  there  are  other 
proofs — direct  proofs — to  substantiate  the  indirect  ones 
embraced  in  this  paper.  The  value  \\-=x  makes  the 
length  of  the  little  line  ££in  the  figure  (2)  -fa\  because 
x— $=f^— %=-£i  and  consequently  the  area  of  the  little 
square  -g^ — altho'  this  area,  as  algebraically  expressed 
by  ^-—6-f,  becomes  zero,  when  the  value  |^  for  x  is 
substituted  in  it — and  the  area  of  the  little  rectangles 
I,  for  ^X  3X*=;f 

A  development  of  the  figure  (2),  on  a  large  scale, 
seems  to  fully  confirm  the  value  ^  for  the  line^rz.  The 
conclusion  therefore  forces  itself  on  us  that  xz-\-x=y  is 
not  a  true  equation — that  jfa  must  be  added  to  the 
second  member  to  make  it  so:  that  is,  that  xz -}-x=^-s\^. 
Now  if  the  first  member  be  squared  by  adding  £-,  the 
second  member,  now  its  equal,  is  also  squared  by  adding 

itoit.  Thus  ^*+/,-+i=p+TVff+i=W-  Extracting 
the  square  root  of  both  members  of  this  equation  we 
have 

*+i=tt  and  *=»-i=H- 

From  all  of  which  it  would  appear  that  for  this  par- 
ticular triangle  the  Pythagorean  law  is  not  true — that  it 
is  an  exception.  Not  only  this,  but  as  there  can  be  con- 
structed an  infinite  number  of  such  triangles,  that  it  is 
only  true  for  what  have  been  rightly  called  the  "  right 
angle  triangle  numbers  ;"  not  true  for  any  case  in  which 
is  involved  the  extraction  of  the  square  root  of  an  im- 
perfect square  to  obtain  the  value  of  what  must  be  a 
positive  line,  not  a  myth  whose  existence  depends  on 
the  number  of  decimals  that  be  allowed  to  constitute 
its  tail. 

I  trust  this  has  been  made  clear  to  you  ;  that  the 
mathematics  sustains  the  a  priori  reasoning:  that  one 
law  of  mathematics  cannot  invalidate  another  equally 
vital  for  the  support  of  truth  and  that  mathematics  even 
may  become  crystalized  into  pttrifactions  of  imper- 
fectly understood  laws  and  thus  passed  from  generation 
to  generation  to  the  detriment  of  truth  and  the  discredit 
of  science. 

Now,  having  reached  this  conclusion  from  perfectly 


20  MATHEMATICAL    MICROBES. 

sound  premises — having  so  weakened  this  cornerstone 
of  mathematics  that  we  see  the  whole  structure  of  the 
science  tottering  to  its  fall,  I  beg  to  vacate  the  position 
of  essayist  for  the  evening  to  take  my  place  as  a  mem- 
ber of  the  Club  to  discuss  the  paper  and  to  discuss  it 
from  the  inside,  as  it  were,  before  I  close. 

In  doing  so  I  want  to  point  out  the  value  of  what  in 
these  days  is  called  "The  higher  criticism."*  Things, 
beliefs,  and  what  not,  have  been  overhauled  by  it,  and 
discredited,  if  not  altogether  relegated  to  the  oblivion 
of  pure  fiction.  Men  are  wiser  now  than  formerly — in 
some  things.  The  higher  criticism  destroys  history, 
converts  facts  into  myths,  unsettles  and  perverts  relig- 
ious beliefs.  Its  syllogisms  are  founded,  not  on  laws 
or  postulates,  fixed  by  mathematical  processes  of  rea- 
soning, but  on  problematical,  half  revealed,  or  totally 
misunderstood  records  of  past  events  and  on  self 
evolved  premises  with  a  basis  of  common  sense.  In 
short,  it  lacks  unassailable  conclusions — it's  defective 
in  that  it  cannot  begin  with  postulates  that  no  man  can 
fail  to  accept,  like  the  postulates  of  mathematics. 

It  is  said  that  figures  will  not  lie — cannot  be  made  to 
lie.  Now,  1  will  reveal  to  you,  that  the  first  part  of  this 
essay  is  a  mathematical  romance,  pure  and  simple  !  But, 
it  was  begun  in  perfect  honesty  and  with  no  other  ob- 
ject in  view  than  the  revelation  of  truth. 

I  believed, — why  I  have  clearly  told  you, — that  all 
these  centuries  a  law  had  been  accepted  as  universal  to 
the  detriment  of  true  science  and  I  set  about  to 
demonstrate  it.  Now,  if  7  can  enter  the  domain  of 
mathematics  and  use  its  own  formulas  so  successfully 
to  prove  that  it  is  wrong,  how  easy  for  those  who  march 
under  the  banner  of  "higher  criticism''  to  assail  ac- 
cepted things  not  based  on  such  well  defined  and  easily 
determined  foundations  as  the  laws  of  mathematics, 
which,  like  those  of  the  Medes  and  Persians,  "  altereth 
not "  ? 

*  The  higher  criticism  meant  is  not  that  which  is  the  true  handmaid  of 
science,  a  fair  sample  of  which  may  be  found  in  a  recent  publication 
"Genesis  and  Modern  Science."— Warren  R.  Perce,  James  Potts  &  Co., 
New  York. 


MATHEMATICAL    MICROBES.  21 

It  seems  to  me  that  what  I  have  done  with  the  higher 
criticism  in  mathematics  would  deceive  the  "  very 
elect."  Now,  I  have  not  only — and  honestly  at  that — 
made  figures  lie,  but  have  brought  into  open  Court  two 
credible  witnesses — viz:  Algebra  and  Geometry — who 
have  sworn  to  the  lies  of  Arithmetic,  thus  endangering 
the  integrity  of  a  venerable  structure,  hoary  with  age 
and  the  Pantheon  of  the  mathematical  Saints  ! 

In  the  science  of  military  engineering  there  are  parts 
or  adjuncts  of  fortifications  and  field  works  called 
chevaux  de  frise  and  trench  cavaliers.  A  teacher  once, 
it  is  said,  in  order,  it  is  presumed,  to  test  a  student's 
knowledge,  asked  him  the  question:  "  Suppose  the  trench 
cavalier  should  run  away,  what  would  you  do  ?  "  With- 
out hesitation  he  answered  "  Mount  a  Cheval  de  frise 
and  go  after  him."  Now,  when  the  Delia  Bacons,  Igna- 
tius Donnellys,  and  others,  like  myself,  for  instance, 
start  out  with  dark  lanterns  to  look  over  ground,  that 
has  been  thoroughly  gone  over  by  others  in  broad  day 
light,  to  search  for  hidden  knowledge,  we  find  stable 
things  in  a  state  of  perturbation.  The  uncertain  light 
presents  things  dimly  ;  outlines  are  not  sharply  defined; 
anything  is  possible  from  the  stampeding  of  the  chevaux 
de  frise  to  the  flight  of  the  trench  cavalier.  The  latter, 
in  this  case,  was  xz-\-x=g.  The  darkness  distorted  it. 
It  was  in  a  state  of  perturbation.  Its  position  was  un- 
certain— it  was,  in  short,  in  a  state  of  flight.  I  could 
neither  catch  it  nor  get  rid  of  it.  It  lurked  in  every 
triangle  and  hid  behind  every  square  encountered  in 
my  pursuit  of  it.  The  persistency  with  which  it  would 
rise,  apparently  from  the  dead,  was  somewhat  appalling. 
When  I  believed  I  was  finally  burying  its  corpse,  no 
sooner  was  the  earth  filled  into  the  grave,  than,  lo  !  there 
were  the  remains  out  of  the  ground  as  before  and  I 
cried  out,  not  for  darkness  and  Bliicher,  but  for  daylight, 
Georg  Cantor  and  Cauchy  ! — whose  acquaintance  you 
will  make  later  on. 

In  a  publication  called  "The  Monist,1'  issue  for  Octo- 
ber, 1896, — Vol.  7,  No.  i,  page  zi —  article,  "The  regen- 
erated logic,''  may  be  found  the  following,  viz: 


22  MATHEMATICAL    MICROBES. 

"It  is  a  remarkable  historical  fact  that  there  is  a 
"branch  of  science  in  which  there  never  has  been  apro- 
"  longed  dispute  concerning  the  proper  objects  of  that 
"  science.  It  is  the  mathematics.  Mistakes  in  mathe- 
"  inatics  occur  not  infrequently,  and  not  being  detected, 
"give  rise  to  false  doctrine,  which  may  continue  a  long 
"  time.  Thus  a  mistake  in  the  evaluation  of  a  definite 
"  integral  by  Laplace,  in  his  '  Mecanique  Ctleste?  led  to 
"an  erroneous  doctrine  about  the  motion  of  the  moon, 
"  which  remained  undetected  for  nearly  a  century.  But 
"  after  the  question  had  once  been  raised,  all  dispute 
"was  brought  to  a  close  within  a  year.  So,  several 
'  demonstrations  in  the  first  book  of  Euclid,  notably 
"  that  in  the  sixteenth  proposition,  are  vitiated  by  the 
"erroneous  assumption  that  a  part  is  necessarily  less 
"than  its  whole.  These  remained  undetected  until 
"  after  the  theory  of  the  non-Euclidean  Geometry  had 
"been  completely  worked  out;  but  since  that  time  no 
"mathematician  has  defended  them,  nor  could  any 
"  competent  mathematician  do  so,  in  view  of  Georg 
"  Cantor's  or  Cauchy's  discoveries." 
Behold  the  Cantor  and  the  Cauchy  !  Great  must  be 
Georg,  or  even  Cauchy,  if  he  has  proved  that  a  part  is 
not  necessarily  less  than  the  whole  of  a  thing  !  The 
audacity  of  my  attacking  the  Pythagorian  law  was 
nothing  to  this  achievement  of  Georg. 

Again,  the  same  publication,  same  number,  subject, 
"Subconscious  Pangeometry,"  page  100,  notices  a  book 
':,  1  from  the  press  of  ^^eubner,  Leipsic,  "  The  theory  of 
parallels."  "A  work  which  perhaps  can  best  be  de- 
"  scribed  as  a  book  on  '  The  Non-Euclidian  Geometry 
"  Inevitable  " — "  It  confers  the  estimable  blessing  on 
"  thinkers  by  giving  them  the  actual  documents  which 
"  are  the  slow,  groping  awakening  of  the  world-mind  at 
"  the  gradual  dawning  of  what  has  now  become  the  full 
"  day  of  self-conscious  non-Euclidian  Geometry." 

Who  knows  but  what  I  am  now  marching  with  the 
"subconscious  "  fellows  under  the  banner  of  Pan-geom- 
etry ?  Who,  according  to  this  notice — it  gives  the  bib- 
liography of  the  siibject — are  proving  that  the  three 


MATHEMATICAL    MICROBES.  -23 

angles  of  a  triangle  are  not  equal  to  two  right  angles; 
that  parallel  lines  are  not  parallel  !  When  I  am  con- 
scious that  I  am  enrolled  with  these  illustrious  fellows 
I  will  again  mount  a  cheval  de  Jrise  and  chase  the  flee- 
ing x*+x=9. 

But,  in  all  seriousness,  what  I  believe  I  have  ac- 
complished is,  that  the  little  line,  kit  of  our  figure  (2)  is 
the  unit  in  this  particular  case,  and  that  I  can  give  in 
any  particular  case  the  unit  and  the  ultimate  atom"  of 
division  —  the  mathematical  microbe  of  exactness.  For 
instance,  without  going  into  details,  this  inconceivable 
quantity  for  xz-}-x=  9  is  122  7*80  ie>  tne  square  root  of 
which  is  sg*04,  and  a  value  (gggf)  determined  for  x,  is 
true  within  -^^  of  the  unit.  When  this  value  is  sub- 
stituted for  x  in  the  equation  xz-\-x=q,  it  gives 
Squaring  the  x  (f-|££)  makes  the 


1  2  2  ^s  Q  i  s  to°  great  for  the  sum  to  be  9. 
Further,  I  can  fix  the  quantities,  areas,  whose  sum 
equals  9,  viz: 

WW  +  tttt=0. 

The  first  is  not  the  square  of  the  second,  and  neither  is 
a  perfect  square,  but  they  are  commensurable;  their 
unit  is  the  same,  viz: 


APPENDIX. 


It  is  presumed  the  geometrical  figures,  explanations, 
etc.,  of  the  foregoing  text  are  sufficiently  comprehended 
to  require  no  repetition.  The  references  below  —  "fig  i'' 
and  "fig  2"  —  are  to  the  geometrical  figures  of  the  text. 

It  may  be  objected  that  some  equations  used  in  what 
follows  —  for  instance  (4)  of  the  essay  —  are  indetermin- 
ate, and  that  the  first  members  reduced  become  identi- 
cal with  the  second.  This  is  true  of  equation  (4),  but 
identity  of  members  insures  their  perfect  equality  to 
start  with,  and  when  a  definite  value  is  given  to  the 
ratio  9^^x  this  identity  disappears  and  the  equation 
ceases  to  be  indeterminate,  and  can  be  true  but  for  one 
value  of  the  variable.  In  this  form  only  is  it  discussed  ; 
it  could  not  be  in  any  other  —  so  with  the  others. 

The  equation  (3)  of  the  essay,  viz.: 

24^3=37^a-f-277-*"—  549     ...     (3) 
as  shown  by  it's  construction,  has  imposed  upon  it  two 
conditions,  that  x3-{-.r=g  and  $£—  6x  shall  equal  zero. 
Hence  two  values,  x=y-^\~i  and  x—  |^  satisfy  it.     The 
first  reduces  it  to  120  4/37  —  336=120  4/37—  336,  the  second 

The  following  equation,  viz:, 

24X3=36x3+2?6x-S4o  )  ,   . 

or,  reducing,  2Xz=^-\-2^x—^      f 

may  be  obtained  from  each  of  these  three; 


which  are  modifications,  authorized  by  the  law,  of 


26  APPENDIX. 


and,  (  **a+4*-3<i  )  (x*—$x+*£)=x*+x-*£  .      .  (b) 

MX"—  202H-26'    v  J 

In  these  equations,^,:-  and  *»!>+*«-36    are  the  num- 

'2!"—  5  4xa—  203+26 

her  of  times,  respectively,  that  the  little  square  /.  n.j.  o. 
is  contained  in  the  sum  of  the  two  rectangles  k.  i.  n.  I 
and  e.  m.  I.  o.  (fig  2),  and  in  this  sumfllus  the  said  square 
—  the  sum  of  the  three  —  viz.  :  x*-\-x—  ^-,  which  the  law 
requires  to  equal  £.  That  the  members  of  equations 
(a)  and  (b)  are  equal  is  apparent;  and,  also,  how  the 
three  preceding  ones  are  obtained  from  them. 

Equation  (4)  is,  like  equation  (3),  remarkable,  in  that 
two  values,  x—v^\~^  and  x—\  satisfy  it.  The  first 
reduces  it  to  101/37  —  28=101/37  —  28,  the  second  to 
J-p-=i|i.  It  is  possible  for  x  to  equal  either  yis]j~1  or 
$£,  but  impossible  for  it  to  equal  %  ;  for  this  value  would 
make  the  line  ki_  (fig.  2)  zero,  and  consequently 
x*—  $x-\-%£-,  zero.  It  would  be  tt&reductio  ad  absurdum 
unless  it  be  interpreted  to  mean  that  the  value  x—\  sat- 
isfies equation  (4)  because  x*—  ^X-\-^-—Q  is,  under  the 
law,  (the  application  of  which  produces  this  impossible 
value),  but  3£—6x=o.  The  law  requires  x—%  to  equal 
/^  —  6x.  If  the  former  be  zero  for  any  value  of  x  the 
latter  must  be  also;  therefore  their  squares  are  equal. 

We  have  then  two  equations,  the  first  members  of 
which  are  identical,  but  the  second  differ  by  x*-\-x—  9; 
thus,  24^3=37^a+277^—  549  .  .  (3) 

6^—  540     ...     (4) 


x*    +   x    —    9 

Any  number  of  times  x*-\-x—c)  added  to  the  second 
member  of  the  first  (3)  will  not  affect  its  value  for 
x—v^\~^.  So  also  any  number  of  times  x^-\-x—g-^ 
added  to  it  will  not  affect  its  value  for  x=%%.  Any 
number  of  times  x*-\-x—  9  added  to  the  second  member 
of  the  latter  (4)  will  not  affect  its  value  for  x=^\~^-\  so 
also  any  number  of  times  x  —  £,  or  x^-^-x—  $£-,  add- 
ed to  it  will  not  affect  its  value  for  x=%.  Which 
is  correct,  x*+x—  zg-=oyx*-\-x—  9=0,  orx*+x—  9^=0; 
are  these  merely  eccentricities  of  cubic  equations? 

Take  the  equation  (4)  of  the  essay,  viz: 


APPENDIX.  27 

A  glance  shows  it  true  for  any  value  of  x,  but  were  the 
ratio     *44/,  known  it  could  be  but  for  one.     Now  the 

O  1       6  4* 

law  requires  this  ratio  to  be  ^J^Vvp  substitute  it  and 
the  equation  becomes 

..-     (5) 


It  would  appear  that  this  equation  must  give  the  value 
for  x  demanded  by  the  law,  for  the  ratio  has  been  given 
the  only  value  it  can  have  under  the  law.  So  it  will,  if 
the  factor  61  —  i^x,  common  to  both  members,  be  divided 
out,  and  in  like  manner  the  equation  will  give  any  value 
for  x.  But  why  not,  with  the  common  factor  left  in? 
No  true  equation  is  affected  by  multiplying  both  mem- 
bers of  it  by  anything:  that  is,  the  equality  of  its  mem- 
bers cannot  be  affected  :  therefore  it  must  follow  that 

(  6  1-8  4*)  144          —  1 

V      5T5      /'i  3-1  8VTT      * 

(which  equation  (5)  becomes  after  dividing  out  tlie  com- 
mon factor)  is  false;  hence  x*-}-x=<),  on  which  it  depends 
for  the  equality  of  its  members,  is  false.  Solved  with 
the  common  factor  left  in,  the  value  *=%%  is  found; 
this  makes  the  common  factor  zero.  This  result  is  con- 
sistent, for  the  members  of  a  false  equation  multiplied 
by  an  expression,  the  value  of  which  is  zero,  reduces 
both  to  zero  and  therefore  to  an  equality. 
Solving  equation  (5). 


3721—  2928*  +  576^  =  4453  —  732  4/37-175^+  288* 

=  732  —  732  4/37 


completing  squares 


—  738-738VTT 


616  '    V  616 

reducing  the  second  member  of  this  last  equation 

138-132VTT_|_/'688+144y5T'\!' 
616  '\  616  ' 

—  -6  16^132-13  8  VTT)    1/6  88  +  1  44  VTT)* 

(516)s  616 

-.421632-481  6  3  8  VTTl    3  4  6  1  4  4+1  6  9  3  4  4  VTT+  1  6  1  8  1  3 
(616)*  (616)* 


28  APPENDIX. 

— 767376-26 2288VTT+ 76787 3 
(676)» 

—  /876-l44VTT\8 
— V 676  / 

TTPTIPP     -K* 1176+288VTT-t-_l_/'688  +  l44VTTV  —  /8  7  6-1  44VTT\a 

•!••*•  dJ.X*C»      *V       ^^  ..    .., *i^  ^^  "' '      '  _ '    I      I    —       '  -~  — 

676  *   \  576  *          ^  676  * 

extracting  square  root  of  both  members 

r 688  +  144  VTT — 876-144VTT 

676 

—  876 1  4  4  VTT 

—  6T6  676 


In  like  manner  any  value,  from  zero  to  infinity,  for  x 
substituted  in  the  ratio,  or,  which  is  the  same  thing,  any 
value  given  to  the  ratio,  will  give  the  value  x=%^. 
One  example  will  suffice:  choosing  a  value  at  random, 
make  6  11-  g4a!=99^=-H£>  ^e  equation  becomes 


(3721  —  2928;tr-|-576.ri)87  =  (6i  —  24^)126 
323727  —  254736^+50  1  12**=  7686—  3024^- 
316041  — 


completing  squares 

x*  _  26  17  1  2  x  I  /I  8686  6)a  =  _  31  6041  _|_.  16839738  73  6 


—  16839738736-16837446692 
(6011  2)s 


extracting  square  root  of  both  members 
x- 


henoe  JIT — i  6i  2+1  ggsee  —  ei 

11C1H-C  -* —  60112  — ^4  • 

Let  the  following  figure  represent  the  little  square 
/  n  j  o  of  fig.  2,  enlarged: 


APPENDIX. 


Now  %^—x  must,  under  the  Pythagorean  law,  have  a 
value—  cannot  be  zero.  Let  ab^  of  this  figure  represent 
—equal  —  it.  cb  being  x—  -|  and  ab  f£—  -*",  £0  will  equal 
.*•—  |—  (-££—  ^)  =  2^r—  3^-,  and  the  square  c  a  e  d, 


A  part  cannot  equal  the  whole;  the  whole  square, 
c  b  fg,  cannot  equal  c  a  e  d,  a  part  of  it,  so  long  as  ab_ 
has  any  value.  But  let  the  expressions  for  the  two 
squares  be  equated  and  the  equation  solved,  thus: 

•     •     •     (6) 


hence 


For  this  vajue,  x—%=-fa,  and  ^ra— 
first  member  of  equation  (6),  hence 


30  APPENDIX. 

thus,   (equation  (6),  )  3*'—  V-^+H^f1^0;  that  is'  the 
first  member  being  the  difference  between  cb*  and  ca*, 

the  difference  is  zero,  and  consequently  -|£—  x=o. 
Now  if  &£—6x=o, 


x*  _  4-JLtr  —  _  11  041  +  8784—  -  _  2 
1  o      "  I  728  1 

**-«*+(**)'=  -HH+«H= 


Let  ^  —  6;r,  the  equal  under  the  law  of  the  second 
member  of  equation  (6),  replace  it,  thus: 


\    (fa)   =        2304~T"23oI:= 


x—  ^1=^.  4/1368,   hence  ^=  gg+^pra,  a  value  a  little 
greater,  as  it  should  be,  than  ^•=yTJ~^>  but  less  than  |^. 

Therefore  -y—  6;tr,  apparently,  is  not  equal  to 
x*—  5-2"+-^-,  and  that  the  latter  is  equal  to  the  square 
of  ca,  a  part  of  itself,  is  because  the  difference 
($x*  —  V-^+Hrt-1)  between  d?  and  ca*  equals  zero,  and 
consequently  fj=o. 

If  -|£  —  x,  &£—6x,  and  x*-\-x—t)-^  equal  zero,  then 
must 

V-6x+(&-xY=o     .     .     .     (c) 

and        — 


Solving  the  first  (  £  )  : 


and  —  fi- 


APPKNDIX.  31 

The    plus   value   is   impossible,    but    the    minus   is 
correct. 

Solving  the  second 


and  *=V&V^fi. 


The  above  appears  to  be  a  cumulative  proof  that 
does  not  equal  9,  and,  so  far  as  algebraic  analysis 
goes,  conclusive.  Opposed  to  it  is  the  geometrical  proof 
embodied  in  the  figure  i  of  the  essay.  If  this,  the 
geometrical  proof,  cannot  be  accepted,  it  would  appear 
that  all  basis  for  mathematical  reasoning  is  removed — 
that  there  is  nothing  to  start  from.  If  the  area  of  a  tri- 
angle is  not  absolutely  equal  to  its  base  multiplied  by 
half  its  altitude,  there  would  appear  to  be  no  basis  for 
determining  the  numerical  value  of  any  surface — no 
basis  in  fact  for  the  algebraic  analysis  above. 

The  proof  of  figure  i  depends,  first,  on  showing  that 
the  square  on  the  hypothenuse  of  a  right  angle  triangle 
equals  exactly  four  times  the  area  of  the  triangle  plus 
the  square  of  the  difference  between  its  base  and  alti- 
tude; and,  secondly,  on  showing  that  the  sum  of  the  areas 
of  the  four  equal  triangles  and  the  square  of  the  said 
difference  equals  the  sum  of  the  squares  of  the  base  and 
altitude.  The  first  is  evident  at  a  glance;  the  second 
requires  but  a  moment's  mental  figuring.  Yet  the 
algebraic  analysis  confirms  in  more  ways  than  one 
the  a  priori  reasoning — the  "common  sense'' — of  the 
essay.  Incommensurability  was  not  admitted,  and, 
consequently  an  impossible  line  for  the  hypothenuse  or 
other  side  of  a  right  angle  triangle.  If  a  triangle  be  a 
triangle  at  all,  its  sides  must  be  positive  and  definite, 
and  therefore  determinable — commensurable  with  some- 
thing in  existence.  This  is  only  another  way  of  stating 
that  one  member  of  an  equation  cannot  be  either  infi- 
nitely less  or  greater  than  the  other — that  the  equality 
of  its  members  must  be  perfect.  It  is  significant  that, 


32  APPENDIX. 

in  the  above  analysis,  whenever  the  equation  is  a  true 
one,  that  which  "squares"  one  member,  always  squares 
the  other. 

If  it  can  be  proved  that  lines  which  m  postulated  to 
be  parallel,  are  not;  that  the  sum  of  the  three  angles  of 
a  triangle  is  not  equal  to  180°;  and  that  "  a  part  is  not 
necessarily  less  than  its  whole,''  then  it  would  appear  not 
so  strange,  even  in  the  face  of  the  simple,  satisfying, 
and  positive  proof  of  the  figure  i,  that  the  Pythagorean 
law  may  be  defective  —  not  general  —  and  only  true  for 
the  "right  angle  triangle  numbers." 

ADDENDUM. 

The  sum  *£-—  6x-\-6.r—  r5=i  (see  page  17)  —  true 
for  any  value  of  x,  and  the  exclusive  creation  of  the 
Pythagorean  law  —  furnishes  an  equation  of  the  2nd  de- 
gree (by  substituting  in  it  for  §£—  6.v  its  equivalent 
(li-|4_2)a  .—ligL-^  an(}  for  the  denominator,  61  —  24^,  of 
the  ratio,  the  value  73—12  4/37,  demanded  by  this  law) 
to  which  the  objections  that  may  be  urged  to  cube 
roots  and  the  equation  (5),  page  27,  (which,  with  the 
factor,  61  —  24^,  common  to  both  members  left  in,  gives 
for  any  value  of  this  ratio  -|£  for  x,  but  y^l~l  when  the 
ratio  is  made  what  the  law  demands  and  the  common 
factor  divided  out)  cannot  apply,  viz: 


-  LLL_:=_l_6r_IH_i 
73-18^31    I  3  —  4 

The  sum,  from  which  this  equation  is  obtained,  is  un- 
questionably equal  to  £,  and  (-SJ-^p~*  Y~si-z±  x  is  unques- 
tionably the  equal  of  ^—6.v,  for  which  it  is  substituted 
in  this  sum.  Thus  no  change  has  been  made  and 
the  sum  is  siill  what  the  Pythagorean  lazv  has  made 
it.  Therefore  the  further  substitution,  in  the  ratio, 

for  61  —  2j^x  the  value  73  —  121/37  which  the  law  re- 
quires, still  further  impresses  on  this  sum  the  condi- 
tions of  the  Pythagorean  law.  and  the  resulting  equa- 
tion —  of  the  2nd  degree,  having  no  common  factor  in 
its  members  and  incapable  of  giving  more  than  one 
value  for  its  unknown  quantity  —  should  give  for  x  the 
value  demanded  by  this  law:  but  it  does  not.  This 
equation  reduces  to 


r2  _  U  1  S_+  3  8  8  V  3Tr-  —  13.  1-  13  2£j 

5"T6  !>  7  c, 

which,  it  is  seen,  is  identical  with  the  equation  on  page 


,_ 

U59 

B86m 


Buf^ington  -  Mathematical  Microbes 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


Aub  1  1  1966 
Jff 


DEC  27  1971 


ST^K 


JUL 
JAN 


Form  L9-39,050-8,'65(F6234s8) 4939 


Form  L9-25m-8, '46  (9852)444 


THE  LIBRARr 

UNIVERSITY  (      LAUFORNIA 
LOS  AI\GELES 


